Best Known (70, 70+179, s)-Nets in Base 4
(70, 70+179, 66)-Net over F4 — Constructive and digital
Digital (70, 249, 66)-net over F4, using
- t-expansion [i] based on digital (49, 249, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(70, 70+179, 105)-Net over F4 — Digital
Digital (70, 249, 105)-net over F4, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 70 and N(F) ≥ 105, using
(70, 70+179, 414)-Net over F4 — Upper bound on s (digital)
There is no digital (70, 249, 415)-net over F4, because
- 3 times m-reduction [i] would yield digital (70, 246, 415)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4246, 415, F4, 176) (dual of [415, 169, 177]-code), but
- residual code [i] would yield OA(470, 238, S4, 44), but
- the linear programming bound shows that M ≥ 120510 936963 900302 789124 723066 073447 088671 101178 269184 854788 285280 658913 691603 030508 274974 720000 / 77708 615139 851578 435610 585674 497674 313378 544338 862201 > 470 [i]
- residual code [i] would yield OA(470, 238, S4, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(4246, 415, F4, 176) (dual of [415, 169, 177]-code), but
(70, 70+179, 468)-Net in Base 4 — Upper bound on s
There is no (70, 249, 469)-net in base 4, because
- 1 times m-reduction [i] would yield (70, 248, 469)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 230880 263021 776730 110163 925408 339771 180517 403155 865725 573314 391409 940359 913457 428924 703846 626103 614614 713784 499367 340249 863251 098698 400150 420295 268480 > 4248 [i]